Page 1 1. Find the nature of roots of the following quadratic equations: (i) 2 2 8 5 0. x x (ii) 2 3 2 6 2 0. x x (iii) 2 5 4 1 0. x x (iv) 5 2 6 0 x x (v) 2 12 4 15 5 0 x x (vi) 2 2 0. x x Sol: (i) The given equation is 2 2 8 5 0. x x This is of the form 2 0, a x b x c where 2, 8 5. a b a n d c Discriminant, 2 2 4 8 4 2 5 64 40 24 0 D b a c Hence, the given equation has real and unequal roots. (ii) The given equation is 2 3 2 6 2 0. x x This is of the form 2 0, a x b x c where 3, 2 6 2. a b a n d c Discriminant, 2 2 4 2 6 4 3 2 24 24 0 D b a c Hence, the given equation has real and equal roots. (iii) The given equation is 2 5 4 1 0. x x This is of the form 2 0, a x b x c where 5, 4 1. a b a n d c Discriminant, 2 2 4 4 4 5 1 16 20 4 0 D b a c Hence, the given equation has no real roots. (iv) The given equation is Page 2 1. Find the nature of roots of the following quadratic equations: (i) 2 2 8 5 0. x x (ii) 2 3 2 6 2 0. x x (iii) 2 5 4 1 0. x x (iv) 5 2 6 0 x x (v) 2 12 4 15 5 0 x x (vi) 2 2 0. x x Sol: (i) The given equation is 2 2 8 5 0. x x This is of the form 2 0, a x b x c where 2, 8 5. a b a n d c Discriminant, 2 2 4 8 4 2 5 64 40 24 0 D b a c Hence, the given equation has real and unequal roots. (ii) The given equation is 2 3 2 6 2 0. x x This is of the form 2 0, a x b x c where 3, 2 6 2. a b a n d c Discriminant, 2 2 4 2 6 4 3 2 24 24 0 D b a c Hence, the given equation has real and equal roots. (iii) The given equation is 2 5 4 1 0. x x This is of the form 2 0, a x b x c where 5, 4 1. a b a n d c Discriminant, 2 2 4 4 4 5 1 16 20 4 0 D b a c Hence, the given equation has no real roots. (iv) The given equation is 2 5 2 6 0 5 10 6 0 x x x x This is of the form 2 0, a x b x c where 5, 10 6. a b a n d c Discriminant, 2 2 4 10 4 5 6 100 120 20 0 D b a c Hence, the given equation has no real roots. (v) The given equation is 2 12 4 15 5 0 x x This is of the form 2 0, a x b x c where 12, 4 15 5. a b a n d c Discriminant, 2 2 4 4 15 4 12 5 240 240 0 D b a c Hence, the given equation has real and equal roots. (vi) The given equation is 2 2 0. x x This is of the form 2 0, a x b x c where 1, 1 2. a b a n d c Discriminant, 2 2 4 1 4 1 2 1 8 7 0 D b a c Hence, the given equation has no real roots. 2. If a and b are distinct real numbers, show that the quadratic equations 2 2 2 2 2 1 0. a b x a b x has no real roots. Sol: The given equation is 2 2 2 2 2 1 0. a b x a b x 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 1 4 2 8 4 8 4 8 8 4 8 4 4 2 4 0 D a b a b a a b b a b a a b b a b a a b b a a b b a b Hence, the given equation has no real roots. 3. Show that the roots of the equation 2 2 0 x p x q are real for all real values of p and q. Sol: Given: 2 2 0 x p x q Here, 2 1, a b p a n d c q Page 3 1. Find the nature of roots of the following quadratic equations: (i) 2 2 8 5 0. x x (ii) 2 3 2 6 2 0. x x (iii) 2 5 4 1 0. x x (iv) 5 2 6 0 x x (v) 2 12 4 15 5 0 x x (vi) 2 2 0. x x Sol: (i) The given equation is 2 2 8 5 0. x x This is of the form 2 0, a x b x c where 2, 8 5. a b a n d c Discriminant, 2 2 4 8 4 2 5 64 40 24 0 D b a c Hence, the given equation has real and unequal roots. (ii) The given equation is 2 3 2 6 2 0. x x This is of the form 2 0, a x b x c where 3, 2 6 2. a b a n d c Discriminant, 2 2 4 2 6 4 3 2 24 24 0 D b a c Hence, the given equation has real and equal roots. (iii) The given equation is 2 5 4 1 0. x x This is of the form 2 0, a x b x c where 5, 4 1. a b a n d c Discriminant, 2 2 4 4 4 5 1 16 20 4 0 D b a c Hence, the given equation has no real roots. (iv) The given equation is 2 5 2 6 0 5 10 6 0 x x x x This is of the form 2 0, a x b x c where 5, 10 6. a b a n d c Discriminant, 2 2 4 10 4 5 6 100 120 20 0 D b a c Hence, the given equation has no real roots. (v) The given equation is 2 12 4 15 5 0 x x This is of the form 2 0, a x b x c where 12, 4 15 5. a b a n d c Discriminant, 2 2 4 4 15 4 12 5 240 240 0 D b a c Hence, the given equation has real and equal roots. (vi) The given equation is 2 2 0. x x This is of the form 2 0, a x b x c where 1, 1 2. a b a n d c Discriminant, 2 2 4 1 4 1 2 1 8 7 0 D b a c Hence, the given equation has no real roots. 2. If a and b are distinct real numbers, show that the quadratic equations 2 2 2 2 2 1 0. a b x a b x has no real roots. Sol: The given equation is 2 2 2 2 2 1 0. a b x a b x 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 1 4 2 8 4 8 4 8 8 4 8 4 4 2 4 0 D a b a b a a b b a b a a b b a b a a b b a a b b a b Hence, the given equation has no real roots. 3. Show that the roots of the equation 2 2 0 x p x q are real for all real values of p and q. Sol: Given: 2 2 0 x p x q Here, 2 1, a b p a n d c q Discriminant D is given by: 2 2 2 2 2 4 4 1 4 0 D b a c p q p q 0 D for all real values of p and q. Thus, the roots of the equation are real. 4. For what values of k are the roots of the quadratic equation 2 3 2 27 0 x k x real and equal? Sol: Given: 2 3 2 27 0 x k x Here, 3, 2 27 a b k a n d c It is given that the roots of the equation are real and equal; therefore, we have: 2 2 0 2 4 3 27 0 4 324 0 D k k 2 2 4 324 81 9 9 9 k k k k o r k 5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and equal. Sol: The given equation is 2 2 5 10 0 2 5 10 0 k x x k x k x This is of the form 2 0, a x b x c where , 2 5 10. a k b k a n d c 2 2 2 4 2 5 4 10 20 40 D b a c k k k k The given equation will have real and equal roots if 0. D Page 4 1. Find the nature of roots of the following quadratic equations: (i) 2 2 8 5 0. x x (ii) 2 3 2 6 2 0. x x (iii) 2 5 4 1 0. x x (iv) 5 2 6 0 x x (v) 2 12 4 15 5 0 x x (vi) 2 2 0. x x Sol: (i) The given equation is 2 2 8 5 0. x x This is of the form 2 0, a x b x c where 2, 8 5. a b a n d c Discriminant, 2 2 4 8 4 2 5 64 40 24 0 D b a c Hence, the given equation has real and unequal roots. (ii) The given equation is 2 3 2 6 2 0. x x This is of the form 2 0, a x b x c where 3, 2 6 2. a b a n d c Discriminant, 2 2 4 2 6 4 3 2 24 24 0 D b a c Hence, the given equation has real and equal roots. (iii) The given equation is 2 5 4 1 0. x x This is of the form 2 0, a x b x c where 5, 4 1. a b a n d c Discriminant, 2 2 4 4 4 5 1 16 20 4 0 D b a c Hence, the given equation has no real roots. (iv) The given equation is 2 5 2 6 0 5 10 6 0 x x x x This is of the form 2 0, a x b x c where 5, 10 6. a b a n d c Discriminant, 2 2 4 10 4 5 6 100 120 20 0 D b a c Hence, the given equation has no real roots. (v) The given equation is 2 12 4 15 5 0 x x This is of the form 2 0, a x b x c where 12, 4 15 5. a b a n d c Discriminant, 2 2 4 4 15 4 12 5 240 240 0 D b a c Hence, the given equation has real and equal roots. (vi) The given equation is 2 2 0. x x This is of the form 2 0, a x b x c where 1, 1 2. a b a n d c Discriminant, 2 2 4 1 4 1 2 1 8 7 0 D b a c Hence, the given equation has no real roots. 2. If a and b are distinct real numbers, show that the quadratic equations 2 2 2 2 2 1 0. a b x a b x has no real roots. Sol: The given equation is 2 2 2 2 2 1 0. a b x a b x 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 1 4 2 8 4 8 4 8 8 4 8 4 4 2 4 0 D a b a b a a b b a b a a b b a b a a b b a a b b a b Hence, the given equation has no real roots. 3. Show that the roots of the equation 2 2 0 x p x q are real for all real values of p and q. Sol: Given: 2 2 0 x p x q Here, 2 1, a b p a n d c q Discriminant D is given by: 2 2 2 2 2 4 4 1 4 0 D b a c p q p q 0 D for all real values of p and q. Thus, the roots of the equation are real. 4. For what values of k are the roots of the quadratic equation 2 3 2 27 0 x k x real and equal? Sol: Given: 2 3 2 27 0 x k x Here, 3, 2 27 a b k a n d c It is given that the roots of the equation are real and equal; therefore, we have: 2 2 0 2 4 3 27 0 4 324 0 D k k 2 2 4 324 81 9 9 9 k k k k o r k 5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and equal. Sol: The given equation is 2 2 5 10 0 2 5 10 0 k x x k x k x This is of the form 2 0, a x b x c where , 2 5 10. a k b k a n d c 2 2 2 4 2 5 4 10 20 40 D b a c k k k k The given equation will have real and equal roots if 0. D 2 20 40 0 20 2 0 0 2 0 0 2 k k k k k o r k k o r k But, for 0, k we get 10 0, which is not true Hence, 2 is the required value of k. 6. For what values of p are the roots of the equation 2 4 3 0. x p x real and equal? Sol: The given equation is 2 4 3 0. x p x This is of the form 2 0, a x b x c where 4, 3. a b p a n d c 2 2 2 4 4 4 3 48 D b a c p p The given equation will have real and equal roots if 0. D 2 2 48 0 48 48 4 3 p p p Hence, 4 3 and 4 3 are the required values of p. 7. Find the nonzero value of k for which the roots of the quadratic equation 2 9 3 0. x k x k are real and equal. Sol: The given equation is 2 9 3 0. x k x k This is of the form 2 0, a x b x c where 9, 3 . a b k a n d c k 2 2 2 4 3 4 9 9 36 D b a c k k k k The given equation will have real and equal roots if 0. D 2 9 36 0 9 4 0 0 4 0 0 4 k k k k k o r k k o r k But, 0 k (Given) Hence, the required values of k is 4. 8. Find the values of k for which the quadratic equation 2 3 1 2 1 1 0. k x k x has real and equal roots. Sol: Page 5 1. Find the nature of roots of the following quadratic equations: (i) 2 2 8 5 0. x x (ii) 2 3 2 6 2 0. x x (iii) 2 5 4 1 0. x x (iv) 5 2 6 0 x x (v) 2 12 4 15 5 0 x x (vi) 2 2 0. x x Sol: (i) The given equation is 2 2 8 5 0. x x This is of the form 2 0, a x b x c where 2, 8 5. a b a n d c Discriminant, 2 2 4 8 4 2 5 64 40 24 0 D b a c Hence, the given equation has real and unequal roots. (ii) The given equation is 2 3 2 6 2 0. x x This is of the form 2 0, a x b x c where 3, 2 6 2. a b a n d c Discriminant, 2 2 4 2 6 4 3 2 24 24 0 D b a c Hence, the given equation has real and equal roots. (iii) The given equation is 2 5 4 1 0. x x This is of the form 2 0, a x b x c where 5, 4 1. a b a n d c Discriminant, 2 2 4 4 4 5 1 16 20 4 0 D b a c Hence, the given equation has no real roots. (iv) The given equation is 2 5 2 6 0 5 10 6 0 x x x x This is of the form 2 0, a x b x c where 5, 10 6. a b a n d c Discriminant, 2 2 4 10 4 5 6 100 120 20 0 D b a c Hence, the given equation has no real roots. (v) The given equation is 2 12 4 15 5 0 x x This is of the form 2 0, a x b x c where 12, 4 15 5. a b a n d c Discriminant, 2 2 4 4 15 4 12 5 240 240 0 D b a c Hence, the given equation has real and equal roots. (vi) The given equation is 2 2 0. x x This is of the form 2 0, a x b x c where 1, 1 2. a b a n d c Discriminant, 2 2 4 1 4 1 2 1 8 7 0 D b a c Hence, the given equation has no real roots. 2. If a and b are distinct real numbers, show that the quadratic equations 2 2 2 2 2 1 0. a b x a b x has no real roots. Sol: The given equation is 2 2 2 2 2 1 0. a b x a b x 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 1 4 2 8 4 8 4 8 8 4 8 4 4 2 4 0 D a b a b a a b b a b a a b b a b a a b b a a b b a b Hence, the given equation has no real roots. 3. Show that the roots of the equation 2 2 0 x p x q are real for all real values of p and q. Sol: Given: 2 2 0 x p x q Here, 2 1, a b p a n d c q Discriminant D is given by: 2 2 2 2 2 4 4 1 4 0 D b a c p q p q 0 D for all real values of p and q. Thus, the roots of the equation are real. 4. For what values of k are the roots of the quadratic equation 2 3 2 27 0 x k x real and equal? Sol: Given: 2 3 2 27 0 x k x Here, 3, 2 27 a b k a n d c It is given that the roots of the equation are real and equal; therefore, we have: 2 2 0 2 4 3 27 0 4 324 0 D k k 2 2 4 324 81 9 9 9 k k k k o r k 5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and equal. Sol: The given equation is 2 2 5 10 0 2 5 10 0 k x x k x k x This is of the form 2 0, a x b x c where , 2 5 10. a k b k a n d c 2 2 2 4 2 5 4 10 20 40 D b a c k k k k The given equation will have real and equal roots if 0. D 2 20 40 0 20 2 0 0 2 0 0 2 k k k k k o r k k o r k But, for 0, k we get 10 0, which is not true Hence, 2 is the required value of k. 6. For what values of p are the roots of the equation 2 4 3 0. x p x real and equal? Sol: The given equation is 2 4 3 0. x p x This is of the form 2 0, a x b x c where 4, 3. a b p a n d c 2 2 2 4 4 4 3 48 D b a c p p The given equation will have real and equal roots if 0. D 2 2 48 0 48 48 4 3 p p p Hence, 4 3 and 4 3 are the required values of p. 7. Find the nonzero value of k for which the roots of the quadratic equation 2 9 3 0. x k x k are real and equal. Sol: The given equation is 2 9 3 0. x k x k This is of the form 2 0, a x b x c where 9, 3 . a b k a n d c k 2 2 2 4 3 4 9 9 36 D b a c k k k k The given equation will have real and equal roots if 0. D 2 9 36 0 9 4 0 0 4 0 0 4 k k k k k o r k k o r k But, 0 k (Given) Hence, the required values of k is 4. 8. Find the values of k for which the quadratic equation 2 3 1 2 1 1 0. k x k x has real and equal roots. Sol: The given equation is 2 3 1 2 1 1 0. k x k x This is of the form 2 0, a x b x c where 3 1, 2 1 1. a k b k a n d c 2 2 2 2 2 4 2 1 4 3 1 1 4 2 1 4 3 1 4 8 4 12 4 4 4 D b ac k k k k k k k k k k The given equation will have real and equal roots if 0. D 2 4 4 0 4 1 0 0 1 0 0 1 k k k k k o r k k o r k Hence, 0 and 1are the required values of k. 9. Find the value of p for which the quadratic equation 2 2 1 7 2 7 3 0. p x p x p has real and equal roots. Sol: The given equation is 2 2 1 7 2 7 3 0. p x p x p This is of the form 2 0, a x b x c where 2 1, 7 2 7 3. a p b p a n d c p 2 4 D b a c 2 7 2 4 2 1 7 3 p p p 2 2 49 28 4 4 14 3 p p p p 2 2 49 28 4 56 4 12 p p p p 2 7 24 16 p p The given equation will have real and equal roots if 0. D 2 7 24 16 0 p p 2 7 24 16 0 p p 2 7 28 4 16 0 p p p 7 4 4 4 0 p p p 4 7 4 0 p p 4 0 7 4 0 p o r pRead More

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